R-groups and geometric structure in the representation theory of SL(N)

Let $F$ be a nonarchimedean local field of characteristic zero and let SL(N) = SL(N,F). This article is devoted to studying the influence of the elliptic representations of SL(N) on the $K$-theory. We provide full arithmetic details. This study reveals an intricate geometric structure. One point of interest is that the $R$-group is realized as an isotropy group. Our results illustrate, in a special case, part (3) of the recent conjecture in [3].